A rigorous program of supervised study designed to expose the student to a particular area of mathematics. Prerequisite: Instructor permission and graduate standing.

Discussion of issues related to the practice of teaching, pedagogical concerns in college level mathematics, and aspects of the responsibilities of a professional mathematician. Credits may not be used towards a Master's degree. Prerequisite: Graduate standing in mathematics.

Studies the basic principles of linear analysis, including spectral theory of compact and selfadjoint operators. Prerequisite: MATH 7340 and 7310, or equivalent.

Studies groups, rings, fields, modules, tensor products, and multilinear functions. Prerequisite: MATH 5651, 5652, or equivalent.

Devoted to chomology theory: cohomology groups, the universal coefficient theorem, the Kunneth formula, cup products, the cohomology ring of manifolds, Poincare duality, and other topics if time permits. Prerequisite: MATH 7800.

Topics include smooth manifolds and functions, tangent bundles and vector fields, embeddings, immersions, transversality, regular values, critical points, degree of maps, differential forms, de Rham cohomology, and connections. Prerequisite: MATH 5310, 5770, or equivalent.

Definition of homotopy groups, homotopy theory of CW complexes, Huriewich theorem and Whitehead's theorem, Eilenberg-Maclane spaces, fibration and cofibration sequences, Postnikov towers, and obstruction theory. Prerequisite: MATH 7800.

Theory of distributions. Sobolev spaces and their properties (trace and embedding theorems). Theory of elliptic equations. Time-dependent partial differential equations: parabolic and hyperbolic equations. Topics in nonlinear partial differential equations. Prerequisites: MATH 7410 and 7250.

Topics in the theory of operators on a Hilbert space and related areas of function theory.

Theory of number fields and local fields, ramification theory, further topics as chosen by instructor.